Potential Fields of Self Intersecting Gielis Curves for Modeling and Generalized Blending Techniques

Abstract

The definition of Gielis curves allows for the representation of self intersecting curves. The analysis and the understanding of these representations is of major interest for the analytical representation of sectors bounded by multiple subsets of curves (or surfaces), as this occurs for instance in many natural objects. We present a construction scheme based on R-functions to build signed potential fields with guaranteed differential properties, such that their zero-set corresponds to the outer, the inner envelop, or combined subparts of the curve. Our framework is designed to allow for the definition of composed domains built upon Boolean operations between several distinct objects or some subpart of self-intersecting curves, but also provides a representation for soft blending techniques in which the traditional Boolean union and intersection become special cases of linear combinations between the objects’ potential fields. Finally, by establishing a connection between R-functions and Lamé curves, we can extend the domain of the p parameter within the $R_p$-function from the set of the even positive numbers to the real numbers strictly greater than 1, i.e. $p \in ]1, +\infty [$.